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# If K is a positive integer, is K the square of an integer?

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Manager
Joined: 10 Oct 2005
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If K is a positive integer, is K the square of an integer? [#permalink]  28 Jan 2007, 07:41
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If K is a positive integer, is K the square of an integer?

(1) K is divisible by 4
(2) K is divisible by exactly 4 different prime numbers

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E)Statements (1) and (2) TOGETHER are NOT sufficient.
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SVP
Joined: 01 May 2006
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(B) for me

Stat1

o If K=4, K = 2^2
o If K=8, K= 2^2 * 2

INSUFF.

Stat2
The answer to the original question is no. If K is the product of exactly 4 prime numbers, we can not have an integer i such that K = i^2.

This is due to the definition of a prime : a prime is not divisable by any integer except 1 and itself.

We have ,for example, K = 2*3*5*7. We could never have K = 2^2 * 3^2.

SUFF.
Senior Manager
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actually, i'd say E is the answer....

certainly st.2 doesn't speak about all factors, but only about prime factors. so the fact that there is 4 different prime factors doesn't rule out the possibility that it is a square. for example 2^2*3^2*5^2*7^2 is a square with exactly 4 different prime factors.
SVP
Joined: 01 May 2006
Posts: 1798
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(E) it is... yes

I misread the question ... Reading K is the product of 4 primes
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